\[ \int \frac {x^4 \left (-2 b+a x^2\right )}{\left (-b+a x^2\right )^2 \sqrt [4]{-b+a x^2+c x^4}} \, dx \]
Optimal antiderivative \[ -\frac {x \left (c \,x^{4}+a \,x^{2}-b \right )^{\frac {3}{4}}}{2 c \left (-a \,x^{2}+b \right )}-\frac {\arctan \left (\frac {c^{\frac {1}{4}} x}{\left (c \,x^{4}+a \,x^{2}-b \right )^{\frac {1}{4}}}\right )}{4 c^{\frac {5}{4}}}-\frac {\arctanh \left (\frac {c^{\frac {1}{4}} x}{\left (c \,x^{4}+a \,x^{2}-b \right )^{\frac {1}{4}}}\right )}{4 c^{\frac {5}{4}}} \]
command
Integrate[(x^4*(-2*b + a*x^2))/((-b + a*x^2)^2*(-b + a*x^2 + c*x^4)^(1/4)),x]
Mathematica 13.1 output
\[ \frac {-\frac {2 \sqrt [4]{c} x \left (-b+a x^2+c x^4\right )^{3/4}}{b-a x^2}-\text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+a x^2+c x^4}}\right )-\tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+a x^2+c x^4}}\right )}{4 c^{5/4}} \]
Mathematica 12.3 output
\[ \int \frac {x^4 \left (-2 b+a x^2\right )}{\left (-b+a x^2\right )^2 \sqrt [4]{-b+a x^2+c x^4}} \, dx \]________________________________________________________________________________________